TheInfoList

OR: In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a field is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
, and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication * Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
are defined and behave as the corresponding operations on
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
and
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s do. A field is thus a fundamental algebraic structure which is widely used in algebra,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, and many other areas of mathematics. The best known fields are the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
s, the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. Most
cryptographic protocol A security protocol (cryptographic protocol or encryption protocol) is an abstract or concrete protocol that performs a security-related function and applies cryptographic methods, often as sequences of cryptographic primitives. A protocol descri ...
s rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and
squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficul ...
cannot be done with a compass and straightedge. Moreover, it shows that
quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
s are, in general, algebraically unsolvable. Fields serve as foundational notions in several mathematical domains. This includes different branches of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
, which is the standard general context for
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
.
Number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s, the siblings of the field of rational numbers, are studied in depth in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
. Function fields can help describe properties of geometric objects.

# Definition

Informally, a field is a set, along with two
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
s defined on that set: an addition operation written as , and a multiplication operation written as , both of which behave similarly as they behave for
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
s and
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s, including the existence of an additive inverse for all elements , and of a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
for every nonzero element . This allows one to also consider the so-called ''inverse'' operations of subtraction, , and division, , by defining: :, :.

## Classic definition

Formally, a field is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
together with two binary operations on called ''addition'' and ''multiplication''. A binary operation on is a mapping , that is, a correspondence that associates with each ordered pair of elements of a uniquely determined element of . The result of the addition of and is called the sum of and , and is denoted . Similarly, the result of the multiplication of and is called the product of and , and is denoted or . These operations are required to satisfy the following properties, referred to as '' field axioms'' (in these axioms, , , and are arbitrary
element Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of o ...
s of the field ): *
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
of addition and multiplication: , and . * Commutativity of addition and multiplication: , and . *
and
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
: there exist two different elements and in such that and . * Additive inverses: for every in , there exists an element in , denoted , called the ''additive inverse'' of , such that . *
Multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
s: for every in , there exists an element in , denoted by or , called the ''multiplicative inverse'' of , such that . *
Distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
of multiplication over addition: . This may be summarized by saying: a field has two operations, called addition and multiplication; it is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition. Even more summarized: a field is a commutative ring where $0 \ne 1$ and all nonzero elements are invertible under multiplication.

## Alternative definition

Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.
Division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there ...
is, by definition, excluded. In order to avoid existential quantifiers, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two nullary operations (the constants and ). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove t ...
and computing. One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two constants and , since and .The a priori twofold use of the symbol "−" for denoting one part of a constant and for the additive inverses is justified by this latter condition.

# Examples

## Rational numbers

Rational numbers have been widely used a long time before the elaboration of the concept of field. They are numbers that can be written as
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, where and are integers, and . The additive inverse of such a fraction is , and the multiplicative inverse (provided that ) is , which can be seen as follows: : $\frac b a \cdot \frac a b = \frac = 1.$ The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows: :

## Real and complex numbers The
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
s , with the usual operations of addition and multiplication, also form a field. The complex numbers consist of expressions : with real, where is the imaginary unit, i.e., a (non-real) number satisfying . Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for . For example, the distributive law enforces : It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the plane, with Cartesian coordinates given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.

## Constructible numbers In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compass and straightedge. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of constructible numbers. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only compass and
straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and ma ...
. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field of rational numbers. The illustration shows the construction of
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
s of constructible numbers, not necessarily contained within . Using the labeling in the illustration, construct the segments , , and a
semicircle In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
over (center at the
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...
), which intersects the
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
line through in a point , at a distance of exactly $h=\sqrt p$ from when has length one. Not all real numbers are constructible. It can be shown that is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a cube with volume 2, another problem posed by the ancient Greeks.

## A field with four elements

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called , , , and . The notation is chosen such that plays the role of the additive identity element (denoted 0 in the axioms above), and is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example, : , which equals , as required by the distributivity. This field is called a finite field with four elements, and is denoted or . The
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
consisting of and (highlighted in red in the tables at the right) is also a field, known as the '' binary field'' or . In the context of computer science and
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
, and are often denoted respectively by ''false'' and ''true'', and the addition is then denoted XOR (exclusive or). In other words, the structure of the binary field is the basic structure that allows computing with
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented ...
s.

# Elementary notions

In this section, denotes an arbitrary field and and are arbitrary elements of .

## Consequences of the definition

One has and . In particular, one may deduce the additive inverse of every element as soon as one knows . If then or must be 0, since, if , then . This means that every field is an integral domain. In addition, the following properties are true for any elements and : : : : : : if

## The additive and the multiplicative group of a field

The axioms of a field imply that it is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
under addition. This group is called the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structur ...
of the field, and is sometimes denoted by when denoting it simply as could be confusing. Similarly, the ''nonzero'' elements of form an abelian group under multiplication, called the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred ...
, and denoted by or just or . A field may thus be defined as set equipped with two operations denoted as an addition and a multiplication such that is an abelian group under addition, is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is distributive over addition.Equivalently, a field is an algebraic structure of type , such that is not defined, and are abelian groups, and ⋅ is distributive over +. Some elementary statements about fields can therefore be obtained by applying general facts of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. For example, the additive and multiplicative inverses and are uniquely determined by . The requirement follows, because 1 is the identity element of a group that does not contain 0. Thus, the trivial ring, consisting of a single element, is not a field. Every finite
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
of the multiplicative group of a field is cyclic (see ).

## Characteristic

In addition to the multiplication of two elements of ''F'', it is possible to define the product of an arbitrary element of by a positive integer to be the -fold sum : (which is an element of .) If there is no positive integer such that :, then is said to have characteristic 0. For example, the field of rational numbers has characteristic 0 since no positive integer is zero. Otherwise, if there ''is'' a positive integer satisfying this equation, the smallest such positive integer can be shown to be a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. It is usually denoted by and the field is said to have characteristic then. For example, the field has characteristic 2 since (in the notation of the above addition table) . If has characteristic , then for all in . This implies that :, since all other binomial coefficients appearing in the
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
are divisible by . Here, ( factors) is the -th power, i.e., the -fold product of the element . Therefore, the Frobenius map : is compatible with the addition in (and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic quite different from fields of characteristic 0.

## Subfields and prime fields

A '' subfield'' of a field is a subset of that is a field with respect to the field operations of . Equivalently is a subset of that contains , and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that , that for all both and are in , and that for all in , both and are in . Field homomorphisms are maps between two fields such that , , and , where and are arbitrary elements of . All field homomorphisms are injective. If is also
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
, it is called an isomorphism (or the fields and are called isomorphic). A field is called a
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive i ...
if it has no proper (i.e., strictly smaller) subfields. Any field contains a prime field. If the characteristic of is (a prime number), the prime field is isomorphic to the finite field introduced below. Otherwise the prime field is isomorphic to .

# Finite fields

''Finite fields'' (also called ''Galois fields'') are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example is a field with four elements. Its subfield is the smallest field, because by definition a field has at least two distinct elements . The simplest finite fields, with prime order, are most directly accessible using
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his b ...
. For a fixed positive integer , arithmetic "modulo " means to work with the numbers : The addition and multiplication on this set are done by performing the operation in question in the set of integers, dividing by and taking the remainder as result. This construction yields a field precisely if is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. For example, taking the prime results in the above-mentioned field . For and more generally, for any composite number (i.e., any number which can be expressed as a product of two strictly smaller natural numbers), is not a field: the product of two non-zero elements is zero since in , which, as was explained above, prevents from being a field. The field with elements ( being prime) constructed in this way is usually denoted by . Every finite field has elements, where is prime and . This statement holds since may be viewed as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
over its prime field. The
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
of this vector space is necessarily finite, say , which implies the asserted statement. A field with elements can be constructed as the splitting field of the
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
:. Such a splitting field is an extension of in which the polynomial has zeros. This means has as many zeros as possible since the degree of is . For , it can be checked case by case using the above multiplication table that all four elements of satisfy the equation , so they are zeros of . By contrast, in , has only two zeros (namely 0 and 1), so does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of ''the'' finite field with elements, denoted by or .

# History

Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations, algebraic number theory, and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. A first step towards the notion of a field was made in 1770 by Joseph-Louis Lagrange, who observed that permuting the zeros of a
cubic polynomial In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
in the expression : (with being a third
root of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
) only yields two values. This way, Lagrange conceptually explained the classical solution method of Scipione del Ferro and François Viète, which proceeds by reducing a cubic equation for an unknown to a quadratic equation for . Together with a similar observation for equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups. Vandermonde, also in 1770, and to a fuller extent, Carl Friedrich Gauss, in his '' Disquisitiones Arithmeticae'' (1801), studied the equation : for a prime and, again using modern language, the resulting cyclic Galois group. Gauss deduced that a regular -gon can be constructed if . Building on Lagrange's work, Paolo Ruffini claimed (1799) that
quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
s (polynomial equations of degree 5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
in 1824.
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as Galois theory today. Both Abel and Galois worked with what is today called an algebraic number field, but conceived neither an explicit notion of a field, nor of a group. In 1871
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
word ''Körper'', which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by . In 1881 Leopold Kronecker defined what he called a ''domain of rationality'', which is a field of rational fractions in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as abstractly as the rational function field . Prior to this, examples of transcendental numbers were known since Joseph Liouville's work in 1844, until
Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermi ...
(1873) and
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficie ...
(1882) proved the transcendence of and , respectively. The first clear definition of an abstract field is due to . In particular, Heinrich Martin Weber's notion included the field F''p''. Giuseppe Veronese (1891) studied the field of formal power series, which led to introduce the field of ''p''-adic numbers. synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections Galois theory, Constructing fields and Elementary notions can be found in Steinitz's work. linked the notion of orderings in a field, and thus the area of analysis, to purely algebraic properties. Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the
primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extens ...
.

# Constructing fields

## Constructing fields from rings

A commutative ring is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses . For example, the integers form a commutative ring, but not a field: the reciprocal of an integer is not itself an integer, unless . In the hierarchy of algebraic structures fields can be characterized as the commutative rings in which every nonzero element is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
(which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
s, and . Fields are also precisely the commutative rings in which is the only
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wit ...
. Given a commutative ring , there are two ways to construct a field related to , i.e., two ways of modifying such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of is , the rationals, while the residue fields of are the finite fields .

### Field of fractions

Given an integral domain , its field of fractions is built with the fractions of two elements of exactly as Q is constructed from the integers. More precisely, the elements of are the fractions where and are in , and . Two fractions and are equal if and only if . The operation on the fractions work exactly as for rational numbers. For example, :$\frac+\frac = \frac.$ It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field. The field of the rational fractions over a field (or an integral domain) is the field of fractions of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
. The field of
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion ...
:$\sum_^\infty a_i x^i \ \left(k \in \Z, a_i \in F\right)$ over a field is the field of fractions of the ring of
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
(in which ). Since any Laurent series is a fraction of a power series divided by a power of (as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.

### Residue fields

In addition to the field of fractions, which embeds injectively into a field, a field can be obtained from a commutative ring by means of a surjective map onto a field . Any field obtained in this way is a
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
, where is a
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
of . If has only one maximal ideal , this field is called the residue field of . The ideal generated by a single polynomial in the polynomial ring (over a field ) is maximal if and only if is irreducible in , i.e., if cannot be expressed as the product of two polynomials in of smaller degree. This yields a field : This field contains an element (namely the residue class of ) which satisfies the equation :. For example, is obtained from by adjoining the imaginary unit symbol , which satisfies , where . Moreover, is irreducible over , which implies that the map that sends a polynomial to yields an isomorphism :

## Constructing fields within a bigger field

Fields can be constructed inside a given bigger container field. Suppose given a field , and a field containing as a subfield. For any element of , there is a smallest subfield of containing and , called the subfield of ''F'' generated by and denoted . The passage from to is referred to by '' adjoining an element'' to . More generally, for a subset , there is a minimal subfield of containing and , denoted by . The compositum of two subfields and of some field is the smallest subfield of containing both and The compositum can be used to construct the biggest subfield of satisfying a certain property, for example the biggest subfield of , which is, in the language introduced below, algebraic over .Further examples include the maximal unramified extension or the maximal abelian extension within .

## Field extensions

The notion of a subfield can also be regarded from the opposite point of view, by referring to being a '' field extension'' (or just extension) of , denoted by :, and read " over ". A basic datum of a field extension is its degree , i.e., the dimension of as an -vector space. It satisfies the formula :. Extensions whose degree is finite are referred to as finite extensions. The extensions and are of degree 2, whereas is an infinite extension.

### Algebraic extensions

A pivotal notion in the study of field extensions are algebraic elements. An element is ''algebraic'' over if it is a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
with coefficients in , that is, if it satisfies a
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
:, with in , and . For example, the imaginary unit in is algebraic over , and even over , since it satisfies the equation :. A field extension in which every element of is algebraic over is called an algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula. The subfield generated by an element , as above, is an algebraic extension of if and only if is an algebraic element. That is to say, if is algebraic, all other elements of are necessarily algebraic as well. Moreover, the degree of the extension , i.e., the dimension of as an -vector space, equals the minimal degree such that there is a polynomial equation involving , as above. If this degree is , then the elements of have the form :$\sum_^ a_k x^k, \ \ a_k \in E.$ For example, the field of Gaussian rationals is the subfield of consisting of all numbers of the form where both and are rational numbers: summands of the form (and similarly for higher exponents) don't have to be considered here, since can be simplified to .

### Transcendence bases

The above-mentioned field of rational fractions , where is an indeterminate, is not an algebraic extension of since there is no polynomial equation with coefficients in whose zero is . Elements, such as , which are not algebraic are called transcendental. Informally speaking, the indeterminate and its powers do not interact with elements of . A similar construction can be carried out with a set of indeterminates, instead of just one. Once again, the field extension discussed above is a key example: if is not algebraic (i.e., is not a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of a polynomial with coefficients in ), then is isomorphic to . This isomorphism is obtained by substituting to in rational fractions. A subset of a field is a transcendence basis if it is algebraically independent (don't satisfy any polynomial relations) over and if is an algebraic extension of . Any field extension has a transcendence basis. Thus, field extensions can be split into ones of the form ( purely transcendental extensions) and algebraic extensions.

## Closure operations

A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
:, with coefficients , has a solution . By the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, is algebraically closed, i.e., ''any'' polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are ''not'' algebraically closed since the equation : does not have any rational or real solution. A field containing is called an ''
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansk ...
'' of if it is algebraic over (roughly speaking, not too big compared to ) and is algebraically closed (big enough to contain solutions of all polynomial equations). By the above, is an algebraic closure of . The situation that the algebraic closure is a finite extension of the field is quite special: by the Artin-Schreier theorem, the degree of this extension is necessarily 2, and is elementarily equivalent to . Such fields are also known as real closed fields. Any field has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as ''the'' algebraic closure and denoted . For example, the algebraic closure of is called the field of algebraic numbers. The field is usually rather implicit since its construction requires the
ultrafilter lemma In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X ( ...
, a set-theoretic axiom that is weaker than the axiom of choice. In this regard, the algebraic closure of , is exceptionally simple. It is the union of the finite fields containing (the ones of order ). For any algebraically closed field of characteristic 0, the algebraic closure of the field of
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion ...
is the field of
Puiseux series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
, obtained by adjoining roots of .

Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.

## Ordered fields

A field ''F'' is called an ''ordered field'' if any two elements can be compared, so that and whenever and . For example, the real numbers form an ordered field, with the usual ordering . The Artin-Schreier theorem states that a field can be ordered if and only if it is a
formally real field In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. Alternative definitions The definition given above i ...
, which means that any quadratic equation :$x_1^2 + x_2^2 + \dots + x_n^2 = 0$ only has the solution . The set of all possible orders on a fixed field is isomorphic to the set of
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
s from the Witt ring of
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s over , to . An Archimedean field is an ordered field such that for each element there exists a finite expression : whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no
infinitesimals In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally ref ...
(elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of . An ordered field is
Dedekind-complete In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...
if all
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an elem ...
s, lower bounds (see
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rat ...
) and limits, which should exist, do exist. More formally, each bounded subset of is required to have a least upper bound. Any complete field is necessarily Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence , every element of which is greater than every infinitesimal, has no limit. Since every proper subfield of the reals also contains such gaps, is the unique complete ordered field, up to isomorphism. Several foundational results in calculus follow directly from this characterization of the reals. The
hyperreals In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of non-standard analysis.

## Topological fields

Another refinement of the notion of a field is a
topological field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is wi ...
, in which the set is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
, such that all operations of the field (addition, multiplication, the maps and ) are
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
s with respect to the topology of the space. The topology of all the fields discussed below is induced from a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
, i.e., a function : that measures a ''distance'' between any two elements of . The completion of is another field in which, informally speaking, the "gaps" in the original field are filled, if there are any. For example, any irrational number , such as , is a "gap" in the rationals in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers , in the sense that distance of and given by the absolute value is as small as desired. The following table lists some examples of this construction. The fourth column shows an example of a zero
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
, i.e., a sequence whose limit (for ) is zero. The field is used in number theory and -adic analysis. The algebraic closure carries a unique norm extending the one on , but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of complex p-adic numbers and is denoted by .

### Local fields

The following topological fields are called '' local fields'':Some authors also consider the fields and to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point that calls them "completely anomalous". * finite extensions of (local fields of characteristic zero) * finite extensions of , the field of Laurent series over (local fields of characteristic ). These two types of local fields share some fundamental similarities. In this relation, the elements and (referred to as uniformizer) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in . (However, since the addition in is done using carrying, which is not the case in , these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper: * Any
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
statement that is true for almost all is also true for almost all . An application of this is the Ax-Kochen theorem describing zeros of homogeneous polynomials in . * Tamely ramified extensions of both fields are in bijection to one another. * Adjoining arbitrary -power roots of (in ), respectively of (in ), yields (infinite) extensions of these fields known as perfectoid fields. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields: $\operatorname \left(\mathbf Q_p \left(p^ \right) \right) \cong \operatorname \left(\mathbf F_p((t))\left(t^\right)\right).$

## Differential fields

Differential fields are fields equipped with a derivation, i.e., allow to take derivatives of elements in the field. For example, the field R(''X''), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory, a variant of Galois theory dealing with
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...
s.

# Galois theory

Galois theory studies algebraic extensions of a field by studying the
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
in the arithmetic operations of addition and multiplication. An important notion in this area is that of finite Galois extensions , which are, by definition, those that are separable and normal. The
primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extens ...
shows that finite separable extensions are necessarily
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
, i.e., of the form :, where is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of are contained in and that has only simple zeros. The latter condition is always satisfied if has characteristic 0. For a finite Galois extension, the Galois group is the group of
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s of that are trivial on (i.e., the bijections that preserve addition and multiplication and that send elements of to themselves). The importance of this group stems from the fundamental theorem of Galois theory, which constructs an explicit
one-to-one correspondence In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the set of
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
s of and the set of intermediate extensions of the extension . By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not solvable (cannot be built from
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s), then the zeros of ''cannot'' be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
s is not solvable for . Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means ...
: : (and ), : (where is regarded as a polynomial in , for some indeterminates , is any field, and ). The
tensor product of fields In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime sub ...
is not usually a field. For example, a finite extension of degree is a Galois extension if and only if there is an isomorphism of -algebras :. This fact is the beginning of
Grothendieck's Galois theory In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in th ...
, a far-reaching extension of Galois theory applicable to algebro-geometric objects.

# Invariants of fields

Basic invariants of a field include the characteristic and the
transcendence degree In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
of over its prime field. The latter is defined as the maximal number of elements in that are algebraically independent over the prime field. Two algebraically closed fields and are isomorphic precisely if these two data agree. This implies that any two
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
algebraically closed fields of the same cardinality and the same characteristic are isomorphic. For example, and are isomorphic (but ''not'' isomorphic as topological fields).

## Model theory of fields

In
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
, a branch of
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, two fields and are called elementarily equivalent if every mathematical statement that is true for is also true for and conversely. The mathematical statements in question are required to be
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
sentences (involving 0, 1, the addition and multiplication). A typical example, for , ''n'' an integer, is : = "any polynomial of degree in has a zero in " The set of such formulas for all expresses that is algebraically closed. The Lefschetz principle states that is elementarily equivalent to any algebraically closed field of characteristic zero. Moreover, any fixed statement holds in if and only if it holds in any algebraically closed field of sufficiently high characteristic. If is an
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
on a set , and is a field for every in , the
ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fa ...
of the with respect to is a field. It is denoted by :, since it behaves in several ways as a limit of the fields : Łoś's theorem states that any first order statement that holds for all but finitely many , also holds for the ultraproduct. Applied to the above sentence , this shows that there is an isomorphismBoth and are algebraically closed by Łoś's theorem. For the same reason, they both have characteristic zero. Finally, they are both uncountable, so that they are isomorphic. :$\operatorname_ \overline \mathbf F_p \cong \mathbf C.$ The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes ) :. In addition, model theory also studies the logical properties of various other types of fields, such as real closed fields or exponential fields (which are equipped with an exponential function ).

## The absolute Galois group

For fields that are not algebraically closed (or not separably closed), the absolute Galois group is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs ''all'' finite separable extensions of . By elementary means, the group can be shown to be the
Prüfer group In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots. ...
, the profinite completion of . This statement subsumes the fact that the only algebraic extensions of are the fields for , and that the Galois groups of these finite extensions are given by :. A description in terms of generators and relations is also known for the Galois groups of -adic number fields (finite extensions of ). Representations of Galois groups and of related groups such as the Weil group are fundamental in many branches of arithmetic, such as the Langlands program. The cohomological study of such representations is done using
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...
. For example, the Brauer group, which is classically defined as the group of central simple -algebras, can be reinterpreted as a Galois cohomology group, namely :.

## K-theory

Milnor K-theory is defined as :$K_n^M\left(F\right) = F^\times \otimes \cdots \otimes F^\times / \left\langle x \otimes \left(1-x\right) \mid x \in F \setminus \ \right\rangle.$ The norm residue isomorphism theorem, proved around 2000 by Vladimir Voevodsky, relates this to Galois cohomology by means of an isomorphism :$K_n^M\left(F\right) / p = H^n\left(F, \mu_l^\right).$ Algebraic K-theory is related to the group of
invertible matrices In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
with coefficients the given field. For example, the process of taking the determinant of an invertible matrix leads to an isomorphism K1(''F'') = ''F''×. Matsumoto's theorem shows that K2(''F'') agrees with K2M(''F''). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.

# Applications

## Linear algebra and commutative algebra

If , then the equation : has a unique solution in a field , namely $x=a^b.$ This immediate consequence of the definition of a field is fundamental in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
. For example, it is an essential ingredient of Gaussian elimination and of the proof that any
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
has a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
. The theory of
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
(the analogue of vector spaces over
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
s instead of fields) is much more complicated, because the above equation may have several or no solutions. In particular systems of linear equations over a ring are much more difficult to solve than in the case of fields, even in the specially simple case of the ring $\Z$ of the integers.

## Finite fields: cryptography and coding theory A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing : ( factors, for an integer ) in a (large) finite field can be performed much more efficiently than the discrete logarithm, which is the inverse operation, i.e., determining the solution to an equation :. In
elliptic curve cryptography Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide ...
, the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve, i.e., the solutions of an equation of the form :. Finite fields are also used in
coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studi ...
and
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
.

## Geometry: field of functions Functions on a suitable
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
into a field can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain: :. This makes these functions a - commutative algebra. For having a ''field'' of functions, one must consider algebras of functions that are integral domains. In this case the ratios of two functions, i.e., expressions of the form :$\frac,$ form a field, called field of functions. This occurs in two main cases. When is a
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
. In this case, one considers the algebra of
holomorphic functions In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex der ...
, i.e., complex differentiable functions. Their ratios form the field of
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. ...
s on . The function field of an algebraic variety (a geometric object defined as the common zeros of polynomial equations) consists of ratios of
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regul ...
s, i.e., ratios of polynomial functions on the variety. The function field of the -dimensional
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
over a field is , i.e., the field consisting of ratios of polynomials in indeterminates. The function field of is the same as the one of any
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
dense subvariety. In other words, the function field is insensitive to replacing by a (slightly) smaller subvariety. The function field is invariant under isomorphism and birational equivalence of varieties. It is therefore an important tool for the study of abstract algebraic varieties and for the classification of algebraic varieties. For example, the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
, which equals the transcendence degree of , is invariant under birational equivalence. For curves (i.e., the dimension is one), the function field is very close to : if is
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
and
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
(the analogue of being compact), can be reconstructed, up to isomorphism, from its field of functions.More precisely, there is an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fr ...
between smooth proper algebraic curves over an algebraically closed field and finite field extensions of .
In higher dimension the function field remembers less, but still decisive information about . The study of function fields and their geometric meaning in higher dimensions is referred to as
birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
. The
minimal model program In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its or ...
attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.

## Number theory: global fields

Global fields are in the limelight in algebraic number theory and arithmetic geometry. They are, by definition,
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s (finite extensions of ) or function fields over (finite extensions of ). As for local fields, these two types of fields share several similar features, even though they are of characteristic 0 and positive characteristic, respectively. This
function field analogy This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of pro ...
can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
concerning the zeros of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ( zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s ...
(open as of 2017) can be regarded as being parallel to the Weil conjectures (proven in 1974 by
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoor ...
). Cyclotomic fields are among the most intensely studied number fields. They are of the form , where is a primitive -th
root of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
, i.e., a complex number satisfying and for all . For being a
regular prime In number theory, a regular prime is a special kind of prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is ...
, Kummer used cyclotomic fields to prove Fermat's Last Theorem, which asserts the non-existence of rational nonzero solutions to the equation :. Local fields are completions of global fields. Ostrowski's theorem asserts that the only completions of , a global field, are the local fields and . Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called the local-global principle. For example, the
Hasse–Minkowski theorem The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent ''locally at all places'', i.e. equivalent over every completion o ...
reduces the problem of finding rational solutions of quadratic equations to solving these equations in and , whose solutions can easily be described. Unlike for local fields, the Galois groups of global fields are not known. Inverse Galois theory studies the (unsolved) problem whether any finite group is the Galois group for some number field . Class field theory describes the abelian extensions, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the Kronecker–Weber theorem, describes the maximal abelian extension of : it is the field : obtained by adjoining all primitive -th roots of unity. Kronecker's Jugendtraum asks for a similarly explicit description of of general number fields . For imaginary quadratic fields, $F=\mathbf Q\left(\sqrt\right)$, , the theory of
complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
describes using elliptic curves. For general number fields, no such explicit description is known.

# Related notions

In addition to the additional structure that fields may enjoy, fields admit various other related notions. Since in any field 0 ≠ 1, any field has at least two elements. Nonetheless, there is a concept of
field with one element In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name ...
, which is suggested to be a limit of the finite fields , as tends to 1. In addition to division rings, there are various other weaker algebraic structures related to fields such as
quasifield In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operation In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...
s, near-fields and
semifield In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. Overview The term semifield has two conflicting meanings, both of which in ...
s. There are also
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
es with field structure, which are sometimes called Fields, with a capital F. The
surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals s ...
s form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The
nimber In mathematics, the nimbers, also called ''Grundy numbers'', are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with ''nimber addition'' and ' ...
s, a concept from game theory, form such a Field as well.

## Division rings Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of a '' division ring'' or ''skew field'';Historically, division rings were sometimes referred to as fields, while fields were called ''commutative fields''. sometimes associativity is weakened as well. The only division rings that are finite-dimensional -vector spaces are itself, (which is a field), and the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s (in which multiplication is non-commutative). This result is known as the Frobenius_theorem_(real_division_algebras), Frobenius theorem. The octonions , for which multiplication is neither commutative nor associative, is a normed Alternative_algebra, alternative division algebra, but is not a division ring. This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor. The non-existence of an odd-dimensional division algebra is more classical. It can be deduced from the hairy ball theorem illustrated at the right.

# References

* * * , especially Chapter 13 * * * * * * . See especially Book 3 () and Book 6 (). * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * {{DEFAULTSORT:Field (Mathematics) Field (mathematics), Algebraic structures Abstract algebra